Title: Milnor Fiber Consistency via Flatness
Date: June 1, 2022
Venue: Iberoamerican Webminar of Young Researchers in Singularity Theory
Abstract: Given a holomorphic family of function germs defining hypersurface singularities, we can ask whether the Milnor fiber varies consistently; in the isolated case, it is well-known that the answer is always yes (in the sense that the family defines a fibration above the complement of the discriminant), and this allows us to obtain a distinguished basis of vanishing cycles for a singularity by perturbing it slightly. In the non-isolated case, this is not always true, and there has long been interest in finding conditions under which this kind of consistency does occur. We give a powerful algebraic condition which is sufficient for this purpose - namely, that the analogous statement will hold so long as the critical locus of the family, considered as an analytic scheme, is flat over the parameter space.
Slides: 20220601_iberoamerican.pdf
Video: https://drive.google.com/file/d/17Swfe-6vWSq0sk8J7l-dBirx6UZJju_G/view
Notes: Basically the same as the previous talk, with the addition of a counterexample due to Bobadilla and some embarrassing technical issues. Watch at your own risk!
Title: Consistency of Milnor Fibers for Deformations of Arbitrary-Dimensional Hypersurface Singularities
Date: May 2, 2022
Venue: University of Wisconsin-Madison Topology and Singularities Seminar
Abstract: Given a holomorphic family of function germs defining hypersurface singularities, we can ask whether the Milnor fiber varies consistently; in the isolated case, it is well-known that the answer is always yes (in the sense that the family defines a fibration above the complement of the discriminant), and this allows us to obtain a distinguished basis of vanishing cycles for a singularity by perturbing it slightly. In the non-isolated case, this is not always true, and there has long been interest in finding conditions under which this kind of consistency does occur. We give a powerful algebraic condition which is sufficient (and possibly necessary) for this purpose - namely, that the analogous statement will hold so long as the critical locus of the family, considered as an analytic scheme, is flat over the parameter space.
Slides: 20220502_uwmadisonsingularities.pdf
Video: https://uwmadison.app.box.com/s/i7kcnfd992qxdky2bat053l3qjjaa8s7/file/953658629482
Notes: It later turned out that the "possibly necessary" part of the abstract was a tad optimistic.
Title: Geometric Intuitions for Flatness
Date: April 7, 2022
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: Flatness is often described as the correct characterization of what it means to have "a nicely varying family of things" in the setting of algebraic geometry. In this talk, which is intended to be pretty low-key, I'll dig a little bit more into what that means, and discuss some theorems and examples that refine and clarify this intuition in various ways.
Title: An Introduction to the Deformation Theory of Complete Intersection Singularities
Date: March 18, 2021
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: Essentially what it says in the title; I'll give a fairly laid-back overview of some of the basic definitions and results about deformations of complete intersection singularities, including the Kodaira-Spencer map and the existence of versal deformations in the isolated case. If time permits, I'll discuss Morsification of isolated singularities. Very little background will be assumed.
Slides: GAGS_CIS.pdf
Notes: Important clarification not mentioned on the slides: The Milnor fiber is well-defined only in certain cases!
Title: Embrace the Singularity: An Introduction to Stratified Morse Theory
Date: April 15, 2020
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: Early on in the semester, Colin told us a bit about Morse Theory, and how it lets us get a handle on the (classical) topology of smooth complex varieties. As we all know, however, not everything in life goes smoothly, and so too in algebraic geometry. Singular varieties, when given the classical topology, are not manifolds, but they can be described in terms of manifolds by means of something called a Whitney stratification. This allows us to develop a version of Morse Theory that applies to singular spaces (and also, with a bit of work, to smooth spaces that fail to be nice in other ways, like non-compact manifolds!), called Stratified Morse Theory. After going through the appropriate definitions and briefly reviewing the results of classical Morse Theory, we'll discuss the so-called Main Theorem of Stratified Morse Theory and survey some of its consequences.
Slides: GAGS_SMT.pdf
Notes: "Colin" here is Colin Crowley. The slides have some blank spaces for pictures I drew during the talk, which didn't end up getting saved anywhere.
Title: Tropicalization Blues
Date: November 13, 2019
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: Tropicalization turns algebro-geometric objects into piecewise linear ones which can then be studied through the lens of combinatorics. In this talk, I will introduce the basic construction, then discuss some of the recent efforts to generalize and improve upon it, touching upon the Giansiracusa tropicalization and developing gazing wistfully in the direction of the machinery of ordered blueprints necessary for the Lorscheid tropicalization.
Notes: Turns out I didn't do so hot estimating the timing on this one, so we didn't get anywhere close to the ordered blueprint stuff. This is a framework due to Oliver Lorscheid which essentially lets one formalize various notions of tropicalization as base changes in an appropriate category; if you'd like to learn more, I wrote this summary for Dan Corey's Introduction to Tropical Geometry course in Spring 2020.
Title: Kindergarten GAGA
Date: April 10, 2019
Venue: University of Wisconsin-Madison Graduate Algebraic Geometry Seminar
Abstract: Join me in regressing to an infantile state as we discuss Serre's 1956 paper Algebraic geometry and analytic geometry, widely considered to be the most influential work ever authored by a mathematician under the age of five. We will define the notion of an analytic space, construct the analytic space associated to any algebraic variety over C, and examine the relationships between the two, including the equivalence between coherent algebraic sheaves and coherent analytic sheaves in the projective case.